Towards a canonical classical natural deduction system

Preprint submitted to Elsevier, 6 July 2012This paper studies a new classical natural deduction system, presented as a typed calculus named lambda-mu- let. It is designed to be isomorphic to Curien and Herbelin's lambda-mu-mu~-calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigot's lambda-mu -calculus with the idea of "coercion calculus" due to Cervesato and Pfenning, accommodating let-expressions in a surprising way: they expand Parigot's syntactic class of named terms. This calculus and the mentioned isomorphism Theta offer three missing components of the proof theory of classical logic: a canonical natural deduction system; a robust process of "read-back" of calculi in the sequent calculus format into natural deduction syntax; a formalization of the usual semantics of the lambda-mu-mu~-calculus, that explains co-terms and cuts as, respectively, contexts and hole- filling instructions. lambda-mu-let is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus; and provides the "read-back" map and the formalized semantics, based on the precise notions of context and "hole-expression" provided by lambda-mu-let. We use "read-back" to achieve a precise connection with Parigot's lambda-mu , and to derive lambda-calculi for call-by-value combining control and let-expressions in a logically founded way. Finally, the semantics , when fully developed, can be inverted at each syntactic category. This development gives us license to see sequent calculus as the semantics of natural deduction; and uncovers a new syntactic concept in lambda-mu-mu~ ("co-context"), with which one can give a new de nition of eta-reduction

Enhanced Realizability Interpretation for Program Extraction

This thesis presents Intuitionistic Fixed Point Logic (IFP), a schema for formal systems aimed to work with program extraction from proofs. IFP in its basic form allows proof construction based on natural deduction inference rules, extended by induction and coinduction. The corresponding system RIFP (IFP with realiz-ers) enables transforming logical proofs into programs utilizing the enhanced re-alizability interpretation. The theoretical research is put into practice in PRAWF1, a Haskell-based proof assistant for program extraction

Relational Parametricity and Control

We study the equational theory of Parigot's second-order λμ-calculus in connection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the target calculus induces a natural notion of equivalence on the λμ-terms. On the other hand, the unconstrained relational parametricity on the λμ-calculus turns out to be inconsistent with this CPS semantics. Following these facts, we propose to formulate the relational parametricity on the λμ-calculus in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc

Canonicity of Proofs in Constructive Modal Logic

In this paper we investigate the Curry-Howard correspondence for constructive modal logic in light of the gap between the proof equivalences enforced by the lambda calculi from the literature and by the recently defined winning strategies for this logic. We define a new lambda-calculus for a minimal constructive modal logic by enriching the calculus from the literature with additional reduction rules and we prove normalization and confluence for our calculus. We then provide a typing system in the style of focused proof systems allowing us to provide a unique proof for each term in normal form, and we use this result to show a one-to-one correspondence between terms in normal form and winning innocent strategies.Comment: Extended version of the TABLEAUX 2023 pape

Relating Justification Logic Modality and Type Theory in Curry–Howard Fashion

This dissertation is a work in the intersection of Justification Logic and Curry--Howard Isomorphism. Justification logic is an umbrella of modal logics of knowledge with explicit evidence. Justification logics have been used to tackle traditional problems in proof theory (in relation to Godel\u27s provability) and philosophy (Gettier examples, Russel\u27s barn paradox). The Curry--Howard Isomorphism or proofs-as-programs is an understanding of logic that places logical studies in conjunction with type theory and -- in current developments -- category theory. The point being that understanding a system as a logic, a typed calculus and, a language of a class of categories constitutes a useful discovery that can have many applications. The applications we will be mainly concerned with are type systems for useful programming language constructs. This work is structured in three parts: The first part is a a bird\u27s eye view into my research topics: intuitionistic logic, justified modality and type theory. The relevant systems are introduced syntactically together with main metatheoretic proof techniques which will be useful in the rest of the thesis. The second part features my main contributions. I will propose a modal type system that extends simple type theory (or, isomorphically, intuitionistic propositional logic) with elements of justification logic and will argue about its computational significance. More specifically, I will show that the obtained calculus characterizes certain computational phenomena related to linking (e.g. module mechanisms, foreign function interfaces) that abound in semantics of modern programming languages. I will present full metatheoretic results obtained for this logic/ calculus utilizing techniques from the first part and will provide proofs in the Appendix. The Appendix contains also information about an implementation of our calculus in the metaprogramming framework Makam. Finally, I conclude this work with a small ``outro\u27\u27, where I informally show that the ideas underlying my contributions can be extended in interesting ways